Question 1

Solve for $$x$$: $$\ln x + \ln(x-3) = \ln 4.$$

Accepted Answer

$$x = 4 or$$ $$x = -1 ($$but only $$x=4 is $$valid)$

Question 2

Find the exact value: $$\sin$$^{-1}$$\left(\sin\left(\frac{8\pi}{7}\right)\right).$$

Accepted Answer

$$\frac{8\pi}{7} - \pi$$

Question 3

Which of the following intervals contains a root of the equation $$x^3 - 3x = 5$ according to the Intermediate Value Theorem?

Accepted Answer

$$(2,3)$$

Question 4

Let $$f(x) = \ln(\frac{1}{3}e$$^{2x}$$ - 3). $$What is the domain of $$f^{-1}(x$$)$$?

Accepted Answer

$$(\ln 3, \infty)$$

Question 5

Compute the following limit: $$\displaystyle \lim_{x 	o 0} \frac{\sin(5x)}{2x}.$$

Accepted Answer

$$\frac{5}{2}$$

Question 6

Suppose $$f(2) = 3$$ and $$f'(2) = 5. $$Evaluate $$\left(\frac{1}{f$$^{-1}$$(x)}\right)' at$$ $$x = 3.$$

Accepted Answer

$$-\frac{1}{15}$$

Question 7

Find all $$x in$$ $$[0,2\pi]$$ such that $$	an^2$ x - 3 = 0.$$

Accepted Answer

$$x = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}$$

Question 8

Let $$f(x) = x^2$ - ax - b$$ for $$x \u003c 1$$ and $$f(x) = \sqrt{ax + b}$$ for $$x \geq 1. $$For $$f(x) to be $$continuous everywhere, what must $$a$$ and $$b$$ satisfy?

Accepted Answer

$$1 - a - b = \sqrt{a + b}$$

Question 9

Which of the following best describes the horizontal asymptote(s) of $$y = \frac{x^3$ + x^2$}{3x^2$ + \sqrt{25x^6$ + x^2$}}$$?

Accepted Answer

$$y = \frac{1}{8}$$ and $$y = -\frac{1}{8}$$

Question 10

Suppose $$f(x) is $$continuous on $$[1,6]$$ and $$f(1) = 1.5$$, $$f(2) = 2.2$$, $$f(3) = 2.9$$, $$f(4) = 3.5$$, $$f(5) = 5.5$$, $$f(6) = 5.9. $$How many solutions does $$f(x) = x$$ have on $$[1,6] by $$the Intermediate Value Theorem?

Accepted Answer

2

Question 11

Let $$f(x) = x \cos(x). $$Where does the tangent line to $$f at$$ $$x = \frac{\pi}{2}$$ intersect the $$y-$$axis?

Accepted Answer

$$y = -1$$

Question 12

Evaluate $$\displaystyle \lim_{h 	o 0} \frac{\ln(1 + 3h)}{7h}.$$

Accepted Answer

$$\frac{3}{7}$$

Question 13

Suppose $$f(x) is $$defined as $$f(x) = x \sin\left(\frac{1}{x}ight)$$ for $$x 
eq 0$$, $$f(0) = 0. Is$$ $$f(x)$$ continuous and differentiable at $$x = 0$$?

Accepted Answer

Continuous but not differentiable at $$x=0$$

Question 14

Express $$\cos(\sin$$^{-1}$$(x/5)) as an $$algebraic function of $$x$$ for $$|x| \leq 5.$$

Accepted Answer

$$\sqrt{1 - (x/5)^2$}$$

Question 15

Let $$f(x) = x^2$ - 3x - 10. $$Compute $$\displaystyle \lim_{x 	o 5^-} \frac{f(x)}{|x-5|}.$$

Accepted Answer

$$-13$$

Question 16

Suppose $$f(x) is $$defined as $$f(x) = x - 2$$ for $$x 
eq 2$$, $$f(2) = 0. Is$$ $$f(x)$$ continuous and differentiable at $$x = 2$$?

Accepted Answer

Not continuous at $$x=2$$

Question 17

Solve for $$x in$$ $$[0,2\pi)$$: $$\sin x = \cos(2x).$$

Accepted Answer

$$x = 0, \frac{2\pi}{3}, \frac{4\pi}{3}$$

Question 18

Suppose $$f(x) is $$continuous and $$f(1) = -2$$, $$f(2) = 3. $$What does the Intermediate Value Theorem guarantee?

Accepted Answer

There is a $$c in$$ $$(1,2)$$ with $$f(c) = 0$$

Question 19

Let $$f(x) = x^2$ - a$$ for $$x \leq 1$$ and $$f(x) = \frac{3x^2$ + 12x - b}{x^2$ + 2x - 3}$$ for $$x \u003e 1. $$For $$f(x) to be $$continuous everywhere, what must $$a$$ and $$b$$ satisfy?

Accepted Answer

$$1 - a = \frac{3 + 12 - b}{1 + 2 - 3}$$

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